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**Hanoi lectures on the arithmetic of hyperelliptic curves.**
*(English)*
Zbl 1294.11107

The paper is an overview of the content in “The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point” [M. Bhargava and B. H. Gross, Studies in Mathematics. Tata Inst. Fundam. Res. 22, 23–91 (2013; Zbl 1303.11072)]. From the introduction:

“Manjul Bhargava and I have recently proved a result on the average order of the \(2\)-Selmer groups of the Jacobians of hyperelliptic curves of fixed genus \(n \geq 1\) over \(\mathbb{Q}\), with a rational Weierstrass point. A surprising fact which emerges is that the average order of this finite group is equal to \(3\), independent of the genus \(n\). This gives us a uniform upper bound of \(\frac{3}{2}\) on the average rank of the Mordell-Weil groups of their Jacobians over \(\mathbb{Q}\). As a consequence, we can use Chabauty’s method to obtain a uniform bound on the number of points on a majority of these curves, when the genus is at least \(2\).”

The paper is expository, and includes brief, but useful introductions to hyperelliptic curves with Weierstrass points, heights of such curves, descent on \(2\)-torsion, and the \(2\)-Selmer group. Afterwards comes a very brief sketch of the proof.

“Manjul Bhargava and I have recently proved a result on the average order of the \(2\)-Selmer groups of the Jacobians of hyperelliptic curves of fixed genus \(n \geq 1\) over \(\mathbb{Q}\), with a rational Weierstrass point. A surprising fact which emerges is that the average order of this finite group is equal to \(3\), independent of the genus \(n\). This gives us a uniform upper bound of \(\frac{3}{2}\) on the average rank of the Mordell-Weil groups of their Jacobians over \(\mathbb{Q}\). As a consequence, we can use Chabauty’s method to obtain a uniform bound on the number of points on a majority of these curves, when the genus is at least \(2\).”

The paper is expository, and includes brief, but useful introductions to hyperelliptic curves with Weierstrass points, heights of such curves, descent on \(2\)-torsion, and the \(2\)-Selmer group. Afterwards comes a very brief sketch of the proof.

Reviewer: Andrew Obus (Charlottesville)

### MSC:

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |

14H25 | Arithmetic ground fields for curves |